In [1]}, [2]}, the authors K. Takasaki and T. Takebe defined the \(\hbar \) -dependent KP hierarchy (\(\hbar \) -KP hierarchy for short) by introduced a formal parameter \(\hbar \) . It is a generalization of the KP hierarchy in the sense that it becomes the KP hierarchy when \(\hbar \rightarrow 1\) . When \(\hbar \rightarrow 0\) , the \(\hbar \) -KP hierarchy becomes the dispersionless KP hierarchy. The \(\hbar \) -KP hierarchy was introduced to study the dispersionless KP hierarchy [1]}. The \(\hbar \) -KP hierarchy is defined by the Lax representation
\(\hbar \frac{\partial L}{\partial x_j}=[B_j,L],\ \ \text{with}\ \ B_j=(L^j)_+,\) 
